World Academy of Science, Engineering and Technology International Journal of Chemical, Molecular, Nuclear, Materials and Metallurgical Engineering Vol:6, No:5, 2012

Modeling the Vapor Pressure of Biodiesel Fuels

International Science Index, Chemical and Molecular Engineering Vol:6, No:5, 2012 waset.org/Publication/2213

O. Castellanos Díaz, F. Schoeggl, H. W. Yarranton, M. A. Satyro, T. M. Lovestead, and T. J. Bruno Abstract—The composition, vapour pressure, and heat capacity of nine biodiesel fuels from different sources were measured. The vapour pressure of the biodiesel fuels is modeled assuming an ideal liquid phase of the fatty acid methyl esters constituting the fuel. New methodologies to calculate the vapour pressure and ideal gas and liquid heat capacities of the biodiesel fuel constituents are proposed. Two alternative optimization scenarios are evaluated: 1) vapour pressure only; 2) vapour pressure constrained with liquid heat capacity. Without physical constraints, significant errors in liquid heat capacity predictions were found whereas the constrained correlation accurately fit both vapour pressure and liquid heat capacity. Keywords—Biodiesel fuels, Fatty acid methyl ester, Heat capacity, Modeling, Vapour pressure I. INTRODUCTION

A

biodiesel fuel is the refined mixture of esters produced by the transesterification of fatty acids from vegetable oil and animal fat (fatty acid methyl esters or FAMEs for short) [1, 2, 3]. These mixtures constitute one of the most promising alternatives for the partial replacement of petroleum-based diesel fuel (petro-diesel). They are renewable, non-mutagenic, non-carcinogenic, biodegradable fuels that can be domestically produced [3, 4]. Biodiesel fuels can be used directly or blended with petroleum diesel, especially lowsulphur fuels, to improve their lubricity without adding any sulphur. These fuels may also improve engine firing because they consist of oxygenated molecules [4]. In order to deploy biodiesel fuels commercially, it is necessary to measure or predict their properties. One important property for the quality control of biodiesel fuels and their blends is volatility, which is directly related to their constituent vapour pressures [2]. For instance, vapour pressure is used to calculate the heat of vaporization in order to compare rates of vaporization and injection characteristics with other fuels. Vapour pressures are also used to assess the cold weather properties of these fuels. Yuan et al. [5] modeled the vapour pressure of three different biodiesel fuels at temperatures above 215 °C using Raoult’s law and the constituent FAME vapour pressures [1, 2]. However, experimental physical properties data for fatty acids and fatty acids methyl esters and biodiesel fuels are scarce and need further development, particularly at lower temperatures.

This paper introduces an improved vapour pressure model applicable for moderate temperatures using an optimization strategy constrained by heat capacity and vapour pressure data. New correlations for FAME heat capacity and vapour pressure are also presented. II. EXPERIMENTAL Compositional, liquid heat capacity, and vapour pressure data for the biodiesel fuels are required to validate the proposed modeling methodology. Table I shows a list of the biodiesel fuels assessed in this research for this purpose, as well as the temperature range of the vapour pressure and heat capacity experimental data. The composition, vapour pressure, and liquid heat capacity of the first nine biodiesel fuels were measured as part of this work. The vapour pressures of the last three biodiesel fuels were obtained from the open literature. A. Biodiesel Fuels Composition The components of each of the biodiesel fuels samples were identified with gas chromatography and mass spectrometry (GC-MS). First, the sample was injected with an automatic sampler into a split/splitless injector set to a 100:1 split ratio. The injector was maintained at a temperature of 350 oC and operated at a constant head pressure of 173 kPa. The stationary phase was a 0.1 µm coating of50 % cyanopropyl-50 % dimethyl polysiloxane, with a temperature program (80 oC for 2 min, 8oC per min to 220 oC, followed by a 220 oC hold for 5 min). This stationary phase provides separations based upon polarity and is specifically intended for the analysis of the FAME compounds that make up biodiesel fuels, and the temperature program is typical for the analysis of such mixtures. Mass spectra were collected and interpreted for each peak from 33 to 750 relative molecular mass (RMM) units [6, 7, 8, 9]. Once the components were identified, the biodiesel fuel samples were analyzed with gas chromatography and flame ionization detection (GC-FID) with external standards to determine mass fraction of each component. FAMEs ranging from C6:0 to C20:1 were identified; with the exception of sample S070717, the majority of each fuel was composed of C18:0, C18:1, and C18:2. Table II summarizes the composition results for the biodiesel fuels listed in Table I. The uncertainty of this data set is approximately 2 %.

O.C.D., F.S., and H.W.Y are with the Department of Chemical and Petroleum Engineering, University of Calgary, AB, Canada (e-mail: hyarrant@ ucalgary.ca). M.A.S is with Virtual Materials Group, Calgary, AB, Canada (e-mail: [email protected]). T.M.L and T.J.B are with the National Institute of Standards and Technology, Boulder, CO, United States (e-mail: [email protected]).

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TABLE I TEMPERATURE RANGE OF VAPOR PRESSURE AND HEAT CAPACITY DATA FOR SELECTED BIODIESEL FUELS Biodiesel fuels Source

Vapour Pressure

Liquid Heat Capacity

Reference

60-196 °C

13-55 °C

This work

-

12-55 °C

This work

140 °C

14-55 °C 10-55 °C

This work This work

80-110 °C 70-100 °C 95-125 °C -

13-55 °C 23-55 °C 10-55 °C 8-55 °C 25-55 °C

This work This work This work This work This work

275-350 °C 215-360 °C 255-340 °C

-

[2] [2] [2]

CB-01 I-25 SB100 MGB100 S102550 S090824 S070717 I26

Sylfat

Certain commercial equipment, materials or supplies are identified in this paper to adequately specify the experimental procedure or description. Such identification does not imply recommendation or endorsement by the National Institute of Standards and Technology, nor does it imply that the equipment, materials or supplies are the best available for the purpose.

B. Biodiesel Fuel Liquid Heat Capacity The liquid heat capacity of the biodiesel fuel samples was measured using a differential scanning calorimeter (DSC) TA Q2000 V24.9 calibrated against indium. The samples were heated at a rate of 5 ºC/min from -40 ºC to 60 ºC and the amount of heat input was recorded. By comparison of the heat flow, the temperature ramp, and the calibration standard, the heat capacity curve of the sample was determined as a function of temperature [10, 11]. The liquid heat capacity is reported at temperatures 2-10 °C above the cloud point to 55 ºC, Table I. C. Biodiesel Fuel Vapor Pressure The vapour pressure of biodiesel fuels was measured using a new static apparatus, Figure 1. The apparatus is designed to perform a series of P-X flashes on a given sample, similar to a differential liberation test. To perform an experiment, the sample vessel is isolated and the rest of the apparatus is placed under a vacuum (the base line pressure) at a pressure below the expected vapour pressure. Then, the sample is opened to the vacuum and the pressure is monitored. Finally, the sample is again isolated and the apparatus is brought back to the base line vacuum. This single flash measurement cycle is repeated as required. An example of the pressure reading for a number of cycles is provided in Figure 2. Samples to be measured may contain lighter impurities. In particular, biodiesel fuels are prone to absorb moisture from the surrounding air [12]. Also, air is always trapped inside the walls of the apparatus when it is exposed to the atmosphere. These impurities may adversely affect the accuracy of the vapour pressure measurement. In order to remove the impurities, several measurement cycles are run as shown in Figure 2. The plot can be divided in three sections: 1) high pressure peaks that are attributed to trapped air; 2) more uniform but decreasing pressure peaks that are attributed to the water and some solvents in the sample; 3) uniform pressure peaks attributed to the vapour pressure of the sample.

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Sections 1 and 2 are known as the degassing section whereas Section 3 is the measurement section.

Fig. 1 chematics of static vapour pressure measurement apparatus; V_01, 02, 03, 04: rubber-sealed in-line valves; TC_01, 02: temperature controllers; PR_01, 02: pressure readers; CT_01, 02: cold traps; vessel: 1/2” inch metal full nipple; pump: turbo-molecular pump 1 0.9

Water + Solvents

Air

0.8

Pressure [kPa x 0.1]

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Canola (South Alberta) Canola (Saskatchewan) Soy (Sunrise, US) Soy (Mountain Gold, US) Rapeseed (Europe) Palm (Europe) Coconut (Europe) Tallow (Alberta) Tallow (South Alberta) Soybean (Idaho) Rapeseed (Idaho) Beef Tallow (Idaho)

Code

Steady Measurement

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 10

20

30

40

50

Time [min]

Fig. 2 Degassing and vapour pressure measurement cycles using apparatus in Figure 1

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TABLE II

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COMPOSITION IN MOLE PERCENTAGE OF FAMES IN SELECTED BIODIESEL FUELS

FAMEs

CB-01

I-25

SB100

MGB100

S102550

S090824

S070717

I26

Sylfat

C6:0 C8:0 C10:0 C12:0 C14:0 C15:0 C16:1 C16:0 C17:0 C17:1 C17:1 C18:0 C18:1(9) C18:1(11) C18:2 C18:3 C20:0 C20:1 MWavg

0 0 0 0 0 0 12.7 12.7 0 0 0 4.1 23.5 1.5 49.9 8.1 0.2 0 291.5

0 0 0 0 0 0 0.9 9.3 0.3 0 0 4.4 57.4 2.8 16.0 7.5 0.4 1.00 293.2

0 0 0 0 0 0 0 11.4 0 0 0 3.2 21.3 1.5 54.9 7.4 0.3 0 291.8

0 0 0 0 0.6 0 0.5 12.5 0 0 0 4.9 27.0 1.6 46.6 6.1 0.3 0 291.3

0 0 0 0 0 0 0 4.8 0 0 0 1.28 59.91 3.68 19.44 9.08 1.26 0.55 294.6

0 0 0 0 1.5 0 0 45.1 0 0 0 3.6 39.5 0 9.8 0.2 0.3 0 283.7

1.0 12.6 7.7 48.3 16.6 0 0 6.7 0 0 0 1.5 4.4 0 1.1 0 0 0 218.2

0 0 0 0 3.4 0.6 2.9 25.6 1.2 0 0 14.8 42.8 1.6 5.8 1.1 0.2 0 286.8

0 0 0 0 0 0 0 10.0 20.2 7.9 7.9 25.3 26.5 2.3 3.6 4.2 0 0 301.3

III. VAPOR PRESSURE MODELING The vapour pressure of the biodiesel fuels is calculated assuming an ideal solution of the constituent FAMEs (Raoult’s law). Calc PBiodiesel = ∑ x j Pj

(1)

j

where x and P are the mole fraction and total ideal vapour pressure of component j, respectively. A correlation is required to determine the vapour pressure of the FAMEs. Rúžička and Majer [13] recommend the Cox equation, among the common vapour pressure equations, to be used when extrapolation is required. This equation has the advantage of not depending on critical properties. In this work, a three degree Cox equation was used [13]:  P   T Re f  (2)  exp (a Pv , 0 + a Pv ,1T + a Pv , 2 T 2 ) ln  V  =  1 +   PRe f  T     where PRef is a reference pressure at TRef, and aPV,1-2-3are the correlation constants. The reference state in the Cox equation should be one close to where the extrapolation is intended. In this case, it is convenient to choose the normal melting point (NMP) of the FAMEs as a reference state. NMP values were obtained from the NIST data base [14]. Note that the vapour pressure of the FAMEs at their NMP is typically unknown and is treated as a fourth adjustable parameter in Equation 2. For substances with high molecular weights such as biodiesel fuels (MW~250 g/mol), it is a challenge to obtain accurate vapour pressure data with which to determine the parameters for the Cox equation. The vapour pressure of these components can be lower than 10-4 kPa at low to moderate temperatures. At these pressure values, the accuracy of direct pressure readings decreases dramatically due to adsorptiondesorption and permeation processes inside the measurement apparatus [15, 16].

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To overcome this issue, indirect measurements such as effusion or transpiration (i.e. gas chromatography) methods are performed [11]. However, these techniques may generate new sources of error coming from the experimental method and/or the processing of the data. An alternative to indirect measurements is to extrapolate accurate vapour pressure data points measured above 10-4 kPa towards lower values. It is advisable to constrain the vapour pressure equation using calorimetric data since these two physical properties are directly related via the ClausiusClapeyron equation and the definition of heat capacity at constant pressure, CP = dH/dT:  d  d ln P j   (3)  ∆C P , j = R  T 2    dT  dT   The vapour pressure correlation can then be constrained as follows [13]:

∑ ln ∑ ∆ , min ln ∆ , (4) where j is the objective function to be minimized, P represents the pressure of the biodiesel fuel, ∆CP is the phase transition heat capacity difference between liquid and vapour phases, KC is a weight factor and i stands for the experimental data points. For an ideal solution assumption, the phase transition heat capacity is given by: ∑! ! ∆ ,! ∆ , (5) To use this method for biodiesel fuels, FAME vapor pressures are required (in this work, experimentally obtained at higher temperatures and extrapolated to lower temperatures) and FAME heat capacities are required that extend to the lower temperatures of interest. IV. FAMES PROPERTIES Eighteen FAMEs ranging in carbon number from 6 to 22 were assessed, as presented in Table III. The vapour pressure data set ranges in temperature from 25 to 300 ºC whereas liquid heat capacity data range from the freezing point to 50

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ºC. Since data were not available for all FAMEs, the modeling approach was developed in four steps: 1) develop a correlation to estimate the phase transition heat capacities; 2) fit the constrained vapour pressure equation to the available vapour pressure data; 3) develop a vapour pressure correlation for FAMEs for which data are not available; 4) predict the vapour pressure of the FAMEs with unavailable data.

1) Ideal Gas Heat Capacity First, the ideal gas heat capacity of the saturated FAMEs with available liquid heat capacity data (Table I) was calculated as follows: ( ,"#$

,%,"#$

) * ,"#$

c CP 0

No data were available for the ideal gas heat capacity of the unsaturated FAMEs. Hence, it is assumed that the departure function from the corresponding saturated FAME is equal to the same departure function as calculated by Joback’s method [17]. The departure function from Joback’s method is given by: 3

is the residual or departure function of the where heat capacity and is calculated using the Peng-Robinson equation of state [17]. Then, the ideal gas heat capacity of these FAMEs was regressed with a second degree polynomial ( +0. ( / (0 1 (0 (8) Figure 3 shows the calculated and regressed ideal gas heat capacity values for methyl caprylate, C10:0 350

i=0

C ( N UC ) = C P0 (0 ) 0 P

330 320 310

i=0

i =0

i=0

3

(12)

3

N CH 3 ∑ a iT i + N CH 2 ∑ biT i + i =0

3

3

i=0

i =0

where NUC is the number of unsaturated bonds (1, 2, or 3), NCH3, NCH2, and NCOO are the number of function groups within the molecule, and a, b, c, d, and e are standard parameters for the method. Equation 12 simplifies to the following expression C P0 ( N UC ) = C P0 (0 )

1 y = 1.22E-03x2 - 2.20E-01x + 2.71E+02 R² = 1.00E+00

280 245

3

N COO ∑ c iT i + N UC ∑ d iT i + ∑ eiT i

N COO ∑ c iT i + ∑ eiT i

300

220

3

1 +    −2   (13)  N  − 6 .1327 x10   UC  + 1 .5493 x10 − 4 T − 1 .842 x10 − 7 T 2     

2) Liquid Heat Capacity Initially, the Dadgostar-Shaw [18] equation was used to calculate the liquid heat capacity of the FAMEs as follows:

Regressed

290

i =0

3

i=0

Calculated 340

3

N CH 3 ∑ a iT i + N CH 2 ∑ biT i +

(7)

CPResidual

Ideal Gas heat Capacity [kJ/kmol.K]

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A. Heat Capacity The phase transition heat capacity is determined as follows: ( ∆ ,"#$ (6) ,%,"#$ ,&,"#$ ' ,%,"#$ ,"#$ 0 where CPL, CPV, and CP stand for the liquid, vapour, and ideal gas heat capacity. Note that since we are concerned with low vapour pressures, the vapour phase can be regarded as ideal and CP,V ≈ CP0 [13]. Hence, correlations for the ideal gas and liquid heat capacity are required.

125 .93 + 1 .453 (10) MW − 344 .176 + 5 .555 x10 − 3 (MW − 344 .176 ) − 0 .1913 = − 8 .863 x10 − 4 (11) MW − 344 .176 −6 − 5 .999 x10 (MW − 344 . 176 )

b CP 0 =

270

295

320

345

370

Temperature [K]

Fig. 3 Calculated and regressed ideal gas heat capacity for methyl caprylate, C10:0

.2 +3-

%

. +3-0

.4 +3-0

(14)

where T is the temperature in Kelvin andα is a similarity variable which is related to the elementary composition of a substance as follows:

3

∑6 56

(15)

∑6 56 $76

The different functions of α are given by:

To generalize Equation 8 for all saturated FAMEs, its parameters were plotted as a function of the molecular mass, Figure 4, and fitted as follows a CP 0 =

− 2 .108 x10 4 + 230 .72 MW − 344 .176 + 0 .625 (MW − 344 .176 )

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.2 +3-

24.5+ 0.34163

2.26713 -

(9)

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(16)

World Academy of Science, Engineering and Technology International Journal of Chemical, Molecular, Nuclear, Materials and Metallurgical Engineering Vol:6, No:5, 2012

TABLE III DATA AVAILABLE FOR SELECTED FAMES AND TEMPERATURE RANGE IN °C [14]

FAME

PV

CPL

Methyl hexanoate Methyl caprylate Methyl caprate Methyl laurate Methyl myristate Methyl pentadecanoate Methyl palmitate Methyl heptadecanoate Methyl stearate Methyl arachidate Methyl behenate Methyl lignocerate Methyl palmitoleate Methyl heptadecenoate Methyl oleate Methyl vaccenate Methyl cis-11-eicosenoate Methyl erucate Methyl linoleate Methyl linolenate

Points

Tmin

Tmax

Points

Tmin

Tmax

65 53 70 112 90 29 110 27 101 29 12 4 33 8 18 12

7.55 33.69 -12.74 -11.00 0.00 21.85 18.00 21.85 21.85 38.00 21.85 26.85 26.85 26.85 26.85 26.85

146.52 145.70 188.20 226.85 237.8 226.85 321.95 226.85 346.95 226.85 258.95 176.85 218.50 176.85 214.95 185.7

12 10 8 7 5 5 5 4 3 -

-33.15 -3.15 6.85 25 26.85 36.85 36.85 46.85 56.85 -

76.85 76.85 76.85 76.85 76.85 76.85 76.85 76.85 76.85 -

C6:0 C8:0 C10:0 C12:0 C14:0 C15:0 C16:0 C17:0 C18:0 C20:0 C22:0 C24:0 C16:1 C17:1 C18:1(11) C18:1(9) C20:1(11) C22:1 C18:2 C18:3

1600 1400

Parameters values

1200

Model

1

1.E-02

0

1.E-02

-1

8.E-03

1000 800

400

-4

200

-5

0

6.E-03

Parameter values Model Regressed

-3

600

cCP,0

-2

bCP,0

aCP,0

4.E-03 2.E-03

-6 150

200

250

300

350

0.E+00 150

200

250 MW

MW

300

350

Fig. 4 Mapping nonlinear data to a higher dimensional feature space

. +3.4 +3-

0.1064 @ 0.38743 9.8231 10CD 4.182 10CE 3

(17) (18)

620 600

Equation 14 was applied to calculate the available liquid heat capacity experimental data of the FAMEs listed in Table 1, with an absolute average relative deviation (AARD) of 2.4%. To improve this accuracy, a modification of Equation 14 is proposed:

( + (9 .509 x10

)

C PL = (2 .279 + a1 (α )) + − 6 . 956 x10 −3 + a 2 (α ) T −6

)

+ a 3 (α ) T 21

CpL [kJ/kmol.K]

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Formula

580 560 540 Experimental Data Dadgostar-Shaw Equation Equation 17

520

(19) 500 300

Figure 5 shows experimental and predicted liquid heat capacity data of methyl palmitate. The FAME specific modification improves the accuracy with an AARD of all of the assessed FAMEs of 0.72%.

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310

320 330 340 Temperature [K]

350

360

Fig. 5 Experimental and predicted liquid heat capacity for C16:0. Data from NIST [14]

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0 Experimental 1.E+01

Constrained

-100

ΔCp [kJ/kmol.K]

Pressure [kPa]

-50

All data

1.E+00 1.E-01

-150

1.E-02

-200

1.E-03

-250

1.E-04

-300

1.E-05

-350

1.E-06 0.002

Experimental All data Constrained

-400 0.00225 0.0025 0.00275 0.003 1/T [1/K]

0.00325 0.0035

250

270

290 310 330 T emperature [K]

350

370

Fig. 6 Experimental and regressed vapor pressure of methyl palmitate (Data from NIST [14])

Experimental "All Data" "Constrained"

60 40

ΔCp [kJ/kg.K]

1.E+01

Pressure [kPa]

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80

1.E+00

20 0 -20 -40

1.E-01 -60

Experimental "All Data" "Constrained"

-80 200

1.E-02 0.0023

0.0025

0.0027 0.0029 1/T [1/K]

0.0031

0.0033

250

300

350

400

Temperature [K]

Fig. 7 Experimental and regressed vapor pressure of methyl caprylate (Data from NIST [14])

B. Vapor Pressure Fitting The Cox equation (Equation 2) was adjusted to the vapour pressure of the FAMEs using the constrained optimization (Equation 4). Two scenarios were evaluated: Scenario 1: “Alldata” is a regression of all the vapour pressure data available with no constraints (KC = 0 in Equation 4); Scenario 2: “Constrained” is a regression of vapour pressure data above 10-4 kPa constrained with liquid heat capacity (a value of KC = 1/100 in Equation 4 scales the heat capacity data in kJ/kmol.K to the same magnitude as the natural log of the vapour pressure data in kPa). Table IV shows the regressed coefficients for both scenarios. Figure 6 shows the results for methyl palmitate. Note that the majority of experimental values below 10-4 kPa were indirect, most of them coming from gas chromatography experiments [14]. Figure 6 shows that both regression scenarios fit the vapour pressures above 10-4 kPa but the constrained fit departs from the data at lower pressures (higher deviations were found with heavier FAMEs). However, the liquid heat capacity values calculated with unconstrained vapour pressure data always deviated from literature data. In addition, in some cases, the heat capacity predicted with the unconstrained equation incorrectly decreased with temperature, as shown in Figure 7 for methyl caprylate.

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The constrained regression produced consistent, accurate predictions of the heat capacity. The AARD values for vapour pressure and heat capacity for all of the FAMEs are 6.7% and 158% in the “All-data” scenario, and 9.0% and 0.7%for the “Constrained” scenario, respectively. Since heat capacity and vapour pressure are related, the “Constrained” correlation is expected to provide a more accurate prediction of the low vapour pressures than the “all data” correlation. C. Vapor Pressure Prediction A new methodology to predict vapour pressure for FAMEs is introduced. For convenience, it is divided into saturated and unsaturated FAMEs vapour pressure. 1) Saturated FAMEs Vapor Pressure Equation In developing the new vapour pressure correlation for saturated FAMEs, the data listed in Table 3 was used as a training set and experimental data of methyl nonadecanoate (C19:0) was used to test the correlation. At a given temperature, 70 ºC in Figure 8, the vapour pressure changes exponentially with the carbon number, as follows:

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Figure 10 (left) shows the experimental data and predictions for the training set. To test Equations 19 to 21, the vapour pressure of C19:0 was predicted, Figure 10 (right). The model predicts the vapour pressure with an AARD of 2.8%.

1.E+00

Presuure [kPa]

1.E-01

2) Unsaturated FAMEs Vapor Pressure Equation For unsaturated FAMEs the amount of experimental data is very small which, in turn, limits the scope of any method based on this data. Data for C18:0, C18:1, C18:2 and C18:3 were used to develop a preliminary correlation; data for C16:1 was used to corroborate the method. Figure 11 shows the experimental data and correlation results for the training set. At high temperatures, any differences among the vapour pressure of the different FAMEs are virtually undistinguishable from the experimental error. However, at low temperatures, the differences become apparent. Therefore, the following departure function is proposed for the unsaturated FAMEs at temperatures below 50 °C (323 K)

1.E-02 1.E-03 1.E-04

Experimental Exponential Regression

1.E-05 1.E-06 0

5

10

15

20

25

Carbon Number

Fig. 8 Mapping nonlinear data to a higher dimensional feature space

.

J,(

expN.

J,2 G

O

(20)

where NC is the carbon number from the fatty acid formula NC:0, and aCN,i are correlation parameters. Each parameter is plotted as a function of the temperature, Figure 9, and fitted as follows: a CN , 0 = 1 .908 exp [0 .01715 T ]

a CN ,1 = − 5 .656 + 0 .02649 T

(22)

− 4 . 5417 x10 − 5 T 2 + 2 .6571 x10 − 8 T 3

(21)

1.E+05

0

a(CN,0) -0.2

Exponential Regression

-0.4 1.E+04

aCN,0

aCN,1

-0.6 -0.8 -1

1.E+03 -1.2

a(CN,1) Polinomic Regression

-1.4 -1.6

1.E+02 260

310

360

410

460

510

560

260

610

Temperature [K]

310

360

410

460

510

560

610

Temperature [K]

Fig. 9 aCN,0and aCN,1parameters in Equation 18 as a function of temperature

1.E+01

1.E+00

Pressure [kPa]

1.E+00

Pressure [kPa]

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F +G -| I

1.E-01 1.E-02 1.E-03 1.E-04

C8:0 Exp.

C10:0 Exp.

C12:0 Exp.

C14:0 Exp.

C16:0 Exp.

C18:0 Exp.

C20:0 Exp.

Training Fit

Experimental Predicted

1.E-01 1.E-02 1.E-03 1.E-04

1.E-05

1.E-05 50

75

100 125 150 175 200 225 250 275 300 Temperature [C]

340 360 380 400 420 440 460 480 500 520 Temperature [K]

Fig. 10 Experimental and predicted values for the training set (left) and the tester, methyl nonadecanoate (right)

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1.E-02 1.E+01 1.E-03

1.E-01

Pressure [kPa]

Pressure [kPa]

1.E+00

1.E-02 1.E-03 C18:0 Exp.

1.E-04

C18:1 Exp.

1.E-05

1.E-05 C18:0 Exp. C18:2 Exp. Training Fit

1.E-06

C18:2 Exp.

1.E-06

1.E-04

C18:3 Exp.

1.E-07

C18:1 Exp. C18:3 Exp.

1.E-07 25

50

75

100 125 150 175 Temperature [C]

200

225

250

20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 Temperature [C]

P +JQR -S P +(-S

.T +GT

1-

/T

QR

JQR U2

1.2

(23)

Experimental Model

1

where NUC is the number of unsaturated double bonds and aUC, bUC, and cUC are the correlation parameters which are related to temperature as follows:

P(Nuc)/P(0)

0.8

a UC = 4 .62 x10 −5 T 2 − 3 .06 x10 −2 T + 5 .05 T ≤ 323 K

0.4

0.2

a UC = 0 T > 3 .23 K bUC = 3 . 39 x10 − 2 T − 9 .93 T ≤ 323 K

0 0

1

2

3

Nuc

(25)

bUC = 0

Fig. 12 Departure function for unsaturated C18 family of FAMEs at 30 °C

T > 3 .23 K c UC = − 2 .97 x10 −2 T + 9 .62 T ≤ 323 K c UC = 0

0.6

(24)

1.E+00

(26)

1.E-01

Pressure [kPa]

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Fig. 11 Experimental and correlated vapour pressure datafor unsaturated C18 family of FAMEs at 30 °C

T > 3 .23 K Figure 12 shows the departure function at 30°C. Note that the C18:2 data point was off the trend at all temperatures. There are very few data points and the outlier may arise from experimental error; more data is required to reach a conclusion. This data point was neglected when fitting Equation 23. The fit to the training data set is shown in Figure 11 (right). To test Equations 23 to 26, the vapour pressure of C16:1 is predicted, Figure 13. The correlation predicts the vapour pressure with an AARD of 2.4%.

1.E-02 1.E-03 1.E-04 C16:1 Experimental data

1.E-05

Prediction 1.E-06 0

50

100 Temperature [C]

150

200

Fig. 13 Experimental and predicted values for methyl palmitoleate

V. BIODIESEL FUEL VAPOR PRESSURE The vapour pressure of the biodiesel fuels listed in Table I was modeled using Raoult’s law (Equation1). Figure 14 shows experimental and predicted vapour pressure data for canola and coconut biodiesel fuels. Two different scenarios were considered, the “All data” and “Constrained” scenarios, depending on which Cox parameters were used for the FAMEs (Table IV). In both cases, Raoult’s Law fits the data well. AARD values are listed in Table V.

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TABLE IV COX EQUATION PARAMETERS FOR ALL-DATA AND CONSTRAINED-DATA SCENARIOS FOR FAMES Scenario 1: All-Data Scenario 2: Constrained-Data

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Formula aPV,1

aPV,2[x103]

aPV,3[x106]

Pref [x106]

aPV,1

aPV,2[x103]

aPV,3[x106]

Pref [x106]

C6:0

3.672

-1.151

1.058

25.833

3.534

-0.642

0.503

3.627

C8:0

2.973

1.854

-2.574

190.05

3.553

-0.694

0.481

108.04

C10:0

3.763

-1.330

1.074

75.905

3.603

-0.707

0.458

84.67

C12:0

5.053

-6.646

6.907

36.558

3.665

-0.776

0.494

69.242

C14:0

4.892

-5.432

5.252

19.399

3.752

-0.914

0.605

35.733

C15:0

3.970

-1.507

1.123

10.787

3.772

-0.758

0.389

12.351

C16:0

4.496

-3.269

2.685

8.846

3.791

-0.796

0.443

14.552

C17:0

4.121

-1.841

1.388

5.095

3.840

-0.883

0.541

6.454

C18:0

4.612

-3.409

2.740

3.297

3.854

-0.792

0.412

5.637

C20:0

3.987

0.102

-1.532

1.107

3.902

-0.866

0.501

3.021

C22:0

4.094

-0.659

-0.412

0.909

4.059

-1.245

0.716

1.695

C16:1**

3.952

0.0834

-0.703

0.000283

4.073

-0.784

0.520

0.000488

C17:1*

3.921

0.0570

-1.384

0.000129

4.110

-0.669

0.350

0.000216

C18:1(9)**

4.242

-0.851

0.255

0.000145

4.288

-1.080

0.527

0.000149

C20:1(11)*

4.397

-1.459

1.070

0.000104

4.153

-0.679

0.381

0.000219

C22:1**

4.457

-1.541

0.995

0.000193

4.299

-1.087

0.647

0.000324

C18:2**

3.982

0.656

-1.324

0.001550a

4.233

-0.855

0.600

0.004187 a

C18:3**

2.552

6.982

-8.629

0.003519 a

4.280

-0.810

0.560

0.000447 a

a. Reference pressure Pref multiplied by 109 TABLE V AVERAGE ABSOLUTE RELATIVE DEVIATION (AATD) PERCENTAGE FOR BIODIESEL FUEL VAPOR PRESSURE AND LIQUID HEAT CAPACITY PREDICTION FAMEs Canola (South Alberta) Canola (Saskatchewan) Soy (Sunrise, US) Soy (Mountain Gold, US) Rapeseed (Europe) Palm (Europe) Coconut (Europe) Tallow (Alberta) Tallow (South Alberta) Soybean (Idaho) Rapeseed (Idaho) Beef Tallow (Idaho) Total

Code CB-01

All Data PV 5.79

CPL 7.26 3.61

I-25 SB100

7.61 4.51

5.91 MGB100 S102550 S090824 S070717 I26 Sylfat

-

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1.53 3.11 12.73

11.17 19.73 40.95 7.67 12.61 12.79

12.30 4.69 6.94 6.62

468

Constrained PV 4.14

9.04 1.23 1.60 9.45

15.57 1.99 6.52 6.19

CPL 2.38 4.20 1.40 0.25 1.26 2.49 0.42 0.68 4.57 1.96

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TABLE VI NOMENCLATURE Units

Adjustable parameter for Equation 4 Phase transition heat capacity Liquid heat capacity Ideal gas heat capacity Residual heat capacity Optimization objective function Molecular mass Number of unsaturated carbon Pressure, Vapour pressure Temperature Reference temperature Mole fraction Similarity function, Equation 12 Stoichiometric value of an element in a compound

kJ/kmol·K kJ/kmol.K kJ/kmol.K kJ/kmol.K Kg/k-mol kPa K -

750 700 650 600 550 500 450 Experimental Canola CB01 Experimental Coconut S170717 Analytical Model - All Data Analytical Model - Constrained

400 350 300 0

5

10

15

20

25

30

35

40

45

50

55

60

Temperature [C]

Fig. 15 Experimental and predicted liquid heat capacity of canola and coconut biodiesel fuels; predictions made with analytical approach

VI. CONCLUSIONS

100 10

Pressure [kPa]

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aPV,i ΔCP CP,L CP0 CPRes J MW NUC P, PV T TRef xi α ν

Definition

Liquid Heat Capacity [kJ/kmol.K]

Symbol

800

1 0.1 0.01 0.001 Experimental Canola CB01 Experimental Coconut S170717 Analytical Model - Constrained Analytical Model - All Data

0.0001 0.00001 0.000001 0

50

100

150

200

250

300

350

400

Temperature [C]

Fig. 14 Experimental and predicted vapour pressure of canola and coconut biodiesel fuel; predictions made with analytical approach (Data from Goodrum [2])

Biodiesel fuel heat capacities were also evaluated, Table V. Since only liquid heat capacity data were available, the predicted liquid heat capacities were calculated for the purpose of comparison as follows:

C P , L , Biodiesels = ∆ C PExp , Biodiesels +

∑C

0 P ,i

,

The vapour pressure of fatty acid methyl esters was modeled using the Cox equation. It was shown that an unconstrained correlation of the vapour pressure may lead to severe deviations in the predicted liquid heat capacity. The constrained correlations acceptably fit both the vapour pressure and liquid heat capacity data. The constrained equation is expected to provide less uncertain predictions of vapour pressure at pressure values close to or below 10-4kPa where reliable vapour pressure experimental data may not be available. New correlations for vapour pressure, liquid heat capacity, and ideal gas heat capacity for FAMEs were also proposed. The vapour pressure and heat capacity of different biodiesel fuels was modeled assuming an ideal solution of FAMEs with an AARD of 6.2 and 2.0%, respectively. ACKNOWLEDGMENT The authors like to acknowledge Shell Canada and the Alberta Research Council for the samples provided. They would also like to thank Dr. Chris Ratcliffe of the Steacie Institute of Molecular Sciences laboratories at the National Research Council of Canada for providing the liquid heat capacity measurements.

(27) REFERENCES

i

[1]

where i stands for the FAMEs that comprised the biodiesel fuels. Figure 15 compares the experimental and predicted (Equation27) liquid heat capacities. The “Constrained” scenario provided more accurate predictions of the heat capacities than the “All data” scenario (Table V). Also, the heat capacity may have an incorrect tendency to decrease with temperature; this behaviour was found to be significant when lighter FAMEs comprise the biodiesel fuels, Table II. Note that the liquid heat capacity was not predicted with the same accuracy as the FAMEs (2% versus 0.7%, Figures 6 and 7). The poorer prediction may result from slightly non-ideal behaviour in the biodiesel fuel liquid phase.

[2] [3]

[4]

[5]

[6] [7]

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Allen, C. A. W.; Watts, K. C.; Ackman, R. G.; and Peg, M. J. Predicting the Viscosity of Biodiesel Fuels from Their Fatty Acid Ester Composition, Fuel, 78, 1999, 1319-1326. Goodrum, J. W. Volatility and Boiling Points of Biodiesel from vegetable Oils and Tallow, Biomass and Bioenergy, 22, 2002, 205-211 Ott, L.; and Bruno, T. Variability of Biodiesel Fuel and Comparison to Petroleum-Derived Diesel Fuel: Application of a Composition and Enthalpy Explicit Distillation Curve Method, Energy & Fuel, 22, 2008,2861-2868 Conceiçao, M. M., Roberlúcia, A. C., Silva, F. C., Bezerra, A. F., Fernandes Jr., V. J., and Souza, A. G. Thermoanalytical Characterization of Castor Oil Biodiesel, Renewable & Sustainable Energy Reviews, 11, 2007, 964-975 Yuan, W., Hansen, A. C.; Zhang, Q. Vapor Pressure and Normal Boiling Point Predictions for Pure Methyl Esters and Biodiesel Fuels, Fuel, 84, 2005,943-950 National Institute of Standards and Technology, NIST, NIST/EPA/NIH Mass Spectral Library Version 1.0, USA, 1995 American Oil Chemist’s Society (AOCS), The Lipid Library, USA, 2001

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World Academy of Science, Engineering and Technology International Journal of Chemical, Molecular, Nuclear, Materials and Metallurgical Engineering Vol:6, No:5, 2012

[8] [9] [10] [11]

[12] [13]

[14]

[15]

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[16] [17] [18]

Bruno, T. J., and Svoronos, P. D. N., CRC Handbook of Fundamental Spectroscopic Correlation Charts, Taylor and Francis Group, 2006 Bruno, T. J., and Svoronos, P. D. N., CRC Handbook of Basic Tables for Chemical Analysis, Third Edition, CRC Press, Boca Raton, 2011 Haines, P. J., Thermal Methods of Analysis: Principles, Applications and Problems, Springer, 2002 Weir, R. D., and de Loos, Th. W. Measurement of the Thermodynamic Properties of Multiple Phases, IUPAC, Physical Chemistry Division, Commission on Thermodynamics, Elsevier, The Netherlands, 2005 Knothe, G.; van Gerpen, J., and Krahl, J., The Biodiesel Handbook, AOCS Press, 2005 Rúžička, K; and Majer, V. Simple and Controlled Extrapolation of Vapor Pressures toward the Triple Point, AIChE J., 42 (6), 1723-1740, 1996 National Institute of Standards and Technology, NIST. ThermoData Engine (TDE) Version 6.0, Pure compounds, Equations of state, Binary mixtures, and Chemical Reactions. NIST Standard reference Database #103b. Thermophysical Research Center. USA. 2011 Roth, A. Vacuum Technology, 3rd edition, North-Holland, Netherlands, 1990 Fulem, M., Private Communication, University of Calgary, Calgary, 2011 Poling, B. E., Prausnitz, J. M., and O’Connell, J. P., The Properties of Gases and Liquids, 5th edition, McGraw-Hill, 2001 Dadgostar, N.; and Shaw, J. A Predictive Correlation for the ConstantPressure Specific Heat Capacity of Pure and Ill-Defined Liquid Hydrocarbons, Fluid Phase Equilibria, Paper in Press, 2011

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